Question

The vertices of a closed convex polygon representing the feasible region of the objective function \(\mathrm{f}\) are \((5,1)(3,5)(4,3)\) and \((2,5)\). Find the maximum value of the function \(\mathrm{f}=8 \mathrm{x}+9 \mathrm{y}\). (1) 61 (2) 69 (3) 59 (4) 49

Short answer

Answer: (2) 69.

Step-by-step solution

Identify the polygon vertices

The vertices of the closed convex polygon are (5,1), (3,5), (4,3) and (2,5).

Write down the objective function

The objective function is given by f(x, y) = 8x + 9y.

Evaluate the function at the vertices

Calculate the value of the function f(x, y) at each vertex of the polygon: (5, 1): f(5, 1) = 8(5) + 9(1) = 40 + 9 = 49 (3, 5): f(3, 5) = 8(3) + 9(5) = 24 + 45 = 69 (4, 3): f(4, 3) = 8(4) + 9(3) = 32 + 27 = 59 (2, 5): f(2, 5) = 8(2) + 9(5) = 16 + 45 = 61

Find the maximum value

Compare the values of the function at the vertices and find the maximum value: 49, 69, 59, 61 The maximum value is 69 which corresponds to vertex (3, 5). So, the maximum value of the function f(x, y) = 8x + 9y is 69, and the correct answer is (2) 69.

Key concepts

Convex Polygon Feasible Region

Let's delve into the concept of a convex polygon feasible region, a fundamental principle in linear programming problems. This geometric figure on the coordinate plane represents a set of all feasible solutions to a linear programming problem. In our exercise, the vertices given, namely \( (5,1), (3,5), (4,3), and (2,5) \), form the corners of this region.

The importance of convexity lies in the fact that if you take any two points within this area and connect them with a straight line, that line will always stay within the region. This particular property is significant because it assures us that the maximum or minimum value of an objective function (in the case of this problem, the maximum) will occur at one of the vertices. Hence, finding the optimum value becomes a matter of evaluating the objective function at these corner points.

Objective Function Evaluation

Moving to the objective function evaluation, this step involves applying a specific mathematical formula - in this case, \( f(x, y) = 8x + 9y \) - to find out which point within our feasible region gives us the highest or lowest value, based on whether we're maximizing or minimizing.

To effectively do this, we substituted the coordinates of each vertex into the function. For instance, for the vertex \( (5,1) \), we calculated \( f(5, 1) = 8(5) + 9(1) \), which resulted in 49. Carrying out this process for all vertices, we are able to identify where the function reaches its peak. It's a crucial step in optimization and requires careful computation to ensure accuracy.

Vertices Calculation

In the process of vertices calculation, it's vital to ensure the precision of identifying the corner points that define the feasible region. Only through accurate locations of these points can the objective function be correctly evaluated. In our example, the coordinates provided were the backing for the entire problem-solving technique.

Determining these vertices mathematically can sometimes involve solving a set of linear equations or inequalities representing the constraints of the linear programming model. In simpler problems, such as the one we tackled, they are provided outright. Each vertex represents a potential solution, and by computing the objective function's value at each of these points, we can discern the optimal solution.

Optimization in Mathematics

Lastly, the concept of optimization in mathematics refers to finding the best possible solution from a set of available alternatives, within the given constraints. It's at the heart of various disciplines such as economics, engineering, and operations research.

In the context of our linear programming problem, optimization entailed searching for the maximum value of the function \( f(x, y) = 8x + 9y \) while adhering to the confines of the convex polygon feasible region. Through methodical evaluation and comparison, the highest function value at the vertices signifies the optimal solution, which helps in making the most effective decision. Recognizing that optimization problems can assume countless forms is key, and the strategies we use, like constructing a feasible region and examining vertices, empower us to address a wide array of real-world scenarios.